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{N ~I- 11 l-i.. ? 9-/, I I
File 629.13 Un 3 as/No. 711
AIR CORPS INFORMATION CIRCULAR
PUBLISHED BY THE CHIEF OF THE AIR CORPS, WASHINGTON. D.C.
Vol. VIII September 15, 1938 No. 711
THE DETERMINATION OF THE PRODUCT OF INERTIA
OF AIRCRAFT CONTROL. SURF ACES
~
(AIRCRAFT BRANCH REPORT )
UNITED STATES
GOVERNMENT PRINTING OFFICE
WASHINGTON : 1938
TABLE OF CONTENTS
Pag~
Summary____ ____ __ ________________________________________________________________ __ ______ _ 1
Object_ _____ ______________________________________________ ___ _________________ ___ _________ __ 1
Discussion____________ ___ ______________________________________________________________ __ ___ _ 1
Definition and physical interpretation of the product of inertia of solid bodies____________________ 1
Development of the formulae for transferring the value of the product of tnertia from one set of
axes to another set of parallel axes ___ _____________________ ______ - - ___ - - - - __ - - - - - - - - __ - - _ _ 2
Significance of static balance as it affects the dynamic balance of control surfaces________________ 2
Product of inertia with respect to two axes that are not mutually perpendicular__ ________________ 2
Product of inertia with respect to two axes that are mutually perpendicular but do not lie in the
same plane ___________________________________________________________________________ " 3
The location of the axes of oscillation to be used for conservative esiimates_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 3
The experimental determination of the product of inertia_ _________ ___ _____ ___ ____ _____________ 4
Conclusions and recommendations______________________________________________________________ 7
Figures
1. Product of inertia of the rudder__ ________________________ _________ _____________________ ___ 8
2. Transferring the value of the product of inertia with respect to one set of axes to another set of
parallel axes___________________________________________________________________________ 8
3. Inclination of the axes for a typical fuselage torsion versus rudder mode of vibration _____________ 9
4. Inclination of the axes for a typical wing bending versus aileron mode of vibration_ ___________ ___ 9
5. Parallel axes for a typical fuselage bending versus rudder mode of vibration_ ________ ____________ 10
6. Axes not in the same plane in the case of a fuselage torsion versus elevator mode of vibration ______ 10
7. Deflection curve of a typical wing in bending___ ____________________________________________ _ 11
8. The three axes used for the experimental determination of the moments of inertia from which the
product of inertia can be calculated __ _______________ ---------------------------------·--· 11
(II)
THE DETERMINATION OF THE PRODUCT OF INERTIA
OF AIRCRAFT CONTROL SURFACES
(Prepared by B. Smilg, Materiel Division, Air Corps, Wright Field, Dayton, Ohio, July 8, 1937)
SUMMARY DISCUSSION
The product of inertia is defined and its physical DEFINITION AND PHYSICAL INTERPRETATION OF THE
significance discussed. The standard formulas for PRODUCT OF INERT IA OF SOLID BODIES
transferring its value from one set of axes to another The product of inertia of a solid body is usually de-set
of axes are derived and the import ance of static fin ed by t he mathematical expression
balance as it affects dynamic balance is pointed out.
The case of the product of inertia with respect to two K ,u= f x y dm __ _____ ___ _____ (!)
axes that are not mutually perpendicular is al so taken where
up. K ,u= product of inertia with respect to the X
It is shown that for conservative estimates the prodand
Y axes
uct of inertia of the aileron and each elevator should be
calculated with respect to the hinge axis of the control
~urfaee and the axis of symmetry of the airplane.
· Similarly, the product of inertia of the rudder should
be calculated with respect to its h inge axis and the
torsional axis of the rear section of the fuselage.
However, in the case of a fuselage with several large
cut-outs in the same cross-section, the product of
inerti;i, of the rudder or elevator should also be calculated
with respect to its hinge line and the bending
axis of. the rear section of the fuselage.
Three methods for determining the product of inertia
experimentally are discussed. These are as follows :
first, by calculations based on the experimental determinations
of the moments of inertia about three
axes lying in the same plane, two of which are mutually
perpendicular; second, by measuring t he torque produced
by a definite angular acceleration; and third, by
the use of auxiliary balance weights to r educe the
product of inertia to zero. In connection with t he
first method, an original scheme for locating the third
axis so as to reduce the effect of unavoidable experimental
errors is developed theoretically, which, when
applied to an actual case, reduced the maximum p ossible
error in the final result from 105 percent to less
than 7 percent.
OBJECT
To indicate the physical significance of the product
of inertia of aircraft control surfaces; to derive tl1e
methods for calculating the product of inertia with
respect to one set of axes, given its value with respect
to a different set of axes; and to discuss various methods
of determining the product of inertia experimentally.
x = x coordinate of the element of mass dm
y = y coordinate of the element of mass dm
In most t exts, the di scussion is limited to the product
of inertia with respect to two mutually perpendicular
axes. In addition, no attempt is usually made
to explain its physical significance. As a result, the
practical application of the product of inertia to the
problems of dynamics is not clearly understood by the
average engineer.
A physical conception of the product of inertia,
especially as it applies to aircraft control surfaces, can
be obtained by a study of the following example.
Figure 1 represents a conventional rudder R of rigid
construction, which is mounted on the pivots P1 and
P2 attached to the fin F, which is assumed to oscillate
as a rigid body about the axis X - X. The hinge line of
the rudder Y- Y and the axis of oscillation X - X will
be assumed to lie in the same plane and to be mutually
perpendicular.
Consider the element of mass dm on the rudder R,
with coordinates x and y, which as a result of the
motion of the fin F and the pivots P1 and P2, is similarly
forced to oscillate about the a xis X - X. Assum ing simple
harmonic motion, then the motion of the element
dm about the X - X axis can be written as
Ox-x=Oo sin wl ___ ____ _____ __ __ (2)
and s= y 0
0 sin wl ____ ______ _____ (3)
where .
Ox-x = angular displacement of the fin about axis
X - X at time l
Oo=maximum arigular displacement of the fin
about axis X - X
s=lin ear di splacement of element dm about
axis X - X at time l
91534-38 (1)
then the linear acceleration will be
d2 -s= - w2 y {}0 sin wt
dt2
and the inertia force on the element dm will be
d2s
dF= (dm) dt2= -w2 y 00 dm sin wt
2
This will cause a torque about the axis Y-Y of an
amount
dTy- y= (x) (dF) = -w2 x y lio dm sin wt
or summing up the torques produced by each element
of mass, then the total torque about the Y-Y axis ·will
be
Ty-y= J-w2 x y /Jo dm sin wt
= Jx y dm(-w2 /Jo sin wt)
However from formula (1) K,.= J x y dm
and from formula (2), since lix- x = lio sin wt
the nd21~Jx-x -- - w 2 Ii o sr. n w t ---------- (3 a )
Thus by substitution,
- - - - - - - - - - - - ( 4)
A study of equation ( 4) shows the physical significance
of the product of inertia, namely, that the product of
inertia of a body with respect to axes X - X and Y- Y, is
equal to the torque induced about the axis Y-Y by a
unit angular acceleration about the axis X-X, or viceversa,
the torque required about the axis X - X to
produce a unit angular acceleration about the axis
Y-Y. This corresponds to the physical interpretation
of the moment of inertia based on the formula
d21Jy_y (
Ty-y=Jy_ydr ------------ 5)
namely that the moment of inertia :>f a body about the
axis Y-Y is equal to the torque about the axis Y-Y
required to produce a unit angular acceleration about
the axis Y-Y.
Thus the general significance of the fact that the
product of inertia of a body with respect to a given pair
of axes is not zero means that an angular acceleration
of that body about either one of these axes will induce
a torque tending to rotate it about the other axis.
DEVELOPMENT OF THE FORMULAE FOR TRANSFERRING
THE VALUE OF THE PRODUCT OF INERTIA FROM ONE
SET OF AXES TO ANOTHER SET OF PARALLEL AXES
It is frequently found necessary to calculate the product
of inertia with respect to one set of axes given the
product of inertia with respect to another set of axes
lying in the same plane. T~e procedure to be followed
can be easily derived from formula (1). Referring to
figure 2, it is desired to express K~2
•2 in terms of Kr
1
vi"
From formula (1),
K,2v
2 = J X2Yz dm
J(,
2
•
2 = J (x1 + Xo) (y1 + Yo) dm
= Jx1y1dm+ fX oY1dm+ J x ,yodm+ fx oyodm
= Kr1•1+ xof Y1dm+ y of X1dm+ x oYofdm
= K ,1.1 + x / Y1M+Yo XiM+xoy oM ________ (6)
where X1 and Y 1 are the x and y coordinates respectively
of the center of gravity with reference to axes
X1- X1 and Y1-Y1
and AI =mass of body .
If the axes X 1- X1 and Yi- Y1 are assumed to pass
through the center of gravity of the body, then
X1 = Y 1=0
so that formula (6) becomes
K ,
2
•
2= (K,,) c.o. + x oYoM - - - - - --- __ (7)
where (K,.) c.g. = product of inertia with respect to the
X and Y axes passing through the center of gravity of
the body.
If in addition, we make either x 0 or y O equal to zero, then
K ,
1
u
2
= K r
2
v1 = (K,v) c , 0• - - - - - - - ___ (8)
SIGNIFICANCE OF STATIC BALAN CE AS IT AFFECTS THE
DYNAMIC BALANCE OF CONTROL SURFACES
The transition from formula (7) to formula (8) indicates
one point that has frequently been overlooked in
the study of dynamic balance, namely the importance
of static balance as it affects the product of inertia.
Comparing the magnitudes of the quantities appearing
in formula (7) and considering axis of Y2-Y2 to be the
hinge axis of the control surface, it is generally found
that (K,. ) c. 0 • is small, whereas Yo , the distance from
the center of. gravity of the control surface to the axis
of oscillation, and M, the mass of the control surface,
are both of relatively large magnitude. Consequently
if the surface is not statically balanced, then XoYoM is
frequently of equal or greater magnitude than (K,.) c.o.
so that the resulting K ,
2
v
2
is large. However, if the sur-face
is statically balanced, then x 0 = 0 so that x 0yoM= O
for all values of Yo and the r esulting product of inertia
is equal to (K ,.) c. 0 . which is generally of small magnitude.
As a corollary, it should be pointed out that in
the case of statically balanced control surfaces, the
product of inertia is independent of the true location
of the axis of oscillation but not of its direction.
PRODUCT OF INERTIA WITH RESPECT TO TW O AXES THAT
ARE NOT MUTUALLY PERPENDICULAR
In studying the product of inertia of aircraft control
surfaces, it is frequently found that the two axes with
respect to which the product of inertia is desired, are
not mutually perpendicular. Cases of this are illustrated
in figures 3 and 4, where 0- 0 is the axis of
oscillation, Y- Y the hinge a xis, <I> the angle between
the 0 - 0 and Y - Y axes in the quadrant where the center
of gravity of the surf ace is located, and X - X the axis
perpendicular to the axis Y-Y, lying in the OY plane,
and passing through the point of intersection of the
0-0 and Y- Y axes.
Referring to figure 3, the product of inertia of the
rudder with respect to the axes 0-0 and Y-Y, is equal
to
IC. = J x r dm
= J x(y sin cJ>-x cos cJ>) dm
=sin cJ> f x y dm-cos cJ> J x2 dm
so that IC.=Kxu sin cJ>-IY-Y cos ,t, __________ _ (9)
Thus IC . is less than IC. , so that if the inclination
of the axes had been neglected, the result would have
been conservative for estimating the critical flutter
speed . However, it is not safe to neglect t his inclination
of the axes in all cases, as can be seen by a study of
figure 4 which represents an aileron-wing combination.
The symbols correspond exactly to those used in
figure 3, and it can be proved similarly that formul:i (9)
is still applicable. However, in this case, cl> is an obtuse
angle so that its cosine is negative in sign. As a result,
Kou might be greater than Kxy especially if I Y-Y were
large compared to Kxu· This condition could exist in
the case of control surfaces which had been balanced
by the addition of several weights. Thus neglecting
the inclination of the axes is not necessarily conservative.
A simple rule for determining whether neglecting the
inclination of t he axes is conservative, is as follows:
consjdering cf> as the angle between the hinge axis and
the axis of oscillation in that quadrant where the center of
gravity of the surface is located, then if <I> is acute, neglecting
the inclination of the axes will a lways be conservative;
if obtuse, the result may be unconservative, especially
if the product of inertia is small compared to t he
moment of inertia.
3
such designs, it appears that the rudder should be statically
balanced at the very least unless provided with an
irreversible control mechanism with a negligible amount
of backlash.
PRODUCT OF INERTIA WI'l'H RESPECT TO TWO AXES THAT
ARE MUTUALLY P E RPENDICULAR BUT DO NOT LIE IN
THE SAME PLANE
The question also sometimes arises with regard to the
product of inertia with respect to two axes that do not
lie in the same plane. As an example, the fuselagetorsion
versus elevator mode of vibration as shown in
figure 6, will be discussed.
Considering the element of mass dm, oscillating with
ilimple harmonic motion about axis X-X, the torsional
axis of the fuselage, then
s=b 00 sin wt
d2s .
dt2= - w2b 00 Slll wt
and the inertia force on the ebment is
dP= -w2b 00 dm sin wt.
However, the component of the ine,tia force that t ends
to rotate the elevator about its hinge line is
dF= (d P)t;= - w2 y 00 dm sin wt
so that as before,
and
K xv= f x'y dm
I(xv=Kx' y
The product of inertia can also be determined with
respect to two axes that are parallel to each other. or
Such cases arise in the wing-torsion versus aileron and
the fuselage-bending versus rudder or elevator modes Thus this proves that the effective product of inertia
of the elevator for the fuselage-torsion versus elevator
mode of vibration is independent of the vertical location
of the torsional axis of the fuselage. Consequently the
product of inertia of the elevator can be calculated with
r espect to its hinge line and the cent er line of the airplane,
regardless of the vertical distance between these
two axes.
of vibration. Using the same nomenclature as in the
previous cases where Y- Y is the hinge line of the control
surface and X-X the axis of oscillation of the body as
shown in fi gure 5, which represents a fuselage bending
versus rudder mode of vibration, then it can be again
proved that
But
Therefore
where
Kxv=fx Y dm
y=x0 + x
Kxu= J x.x dm+ J x2 dm
ICu=XoXoM + I Y-Y
x0 is the distance between the two axes
X. is the distance of the center of gravity of the
control surface aft of the hinge line
JY[ is the mass of the control surface
I Y- y is the moment of inertia of the control
surface about the hinge line.
It is thus obvious that to make K equal to zero for this
mode of vibrat ion, X0 must be negative; that is the
center of gravity of the control surface must lie forward
of the hinge line.
This fuselage bending versus rudder mode of vibration
is likely to occur in airplanes where the rear section
of the fuselage has several large cut-outs lying in the
same cross-section and whose rudder has a low natural
frequency due to spring couplin~ or other devi<;es. For
THE LOCATION OF THE AXES OF OSCILLATION TO BE USED
FOR CONSERVATIVE ESTIMATES
In the preceding discussion, it was assumed that the
control surfaces and the struct ures by which they were
supported, vibrated as rigid bodies about definite axes
of oscillation. Practical aircraft structures, however,
are far from rigid and in addition, the location of the
true axes of oscillation is frequently difficult to determine.
The only accurate means for locating the axes of
oscillation is to measure the vibration amplitudes of the
supporting structure in flight at speeds approaching
the critical flutter speed. The accuracy of other methods
such as those based on the deflection curve of the
wing or fixed tail surface under various types of loading,
is extremely dubious because of the large di screpancies
that may exist between the assumed and the actual
mod~s Qf v~bration, This is :particula,rly true in the
4
case of externally braced structures as well as cantilever
structures that are not very rigid torsionally. However,
foi· cantilever wings that are of great torsional
rigidity, the only practical mode of vibration that can
exist is that of the wing in pure bending. This condition
is satisfied by most stressed-skin metal covered wings.
Consequently it appears that for this type of structure,
the mode of vibration of the wing in bending may
be estimated from the deflection cu rve of the wing
under a load vary ing more or less un iformly from zero
near the root to a maximum at the t ip, corresponding
to the inertia and aerodynamic damping loads set up
by small vibrations of the wing in bending.
In general, it will be found that this deflection curve
is not a straight line and the problem then arises as to
how to estimate the location of the axis of oscillation
given such a deflection curve. To show how this
should be done, an example of a wing-aileron mode of
vibration with a deflection curve (exaggerated) as
shown in figure 7 will be discussed.
If z i_s the di splacement of any point under the loading
described above, then if we assume that t he structure is
vibrating in simple harmonic motion with a mode of
vibration corresponding to this deflect ion curve, then
the motion s of an element of mass drn located on the
aileron can be written as
s=z sin wt
and its acceleration as ·
d
2
s = -w2 z sin wt
dt2
Consequently the inertia force on the element drn will be
d2s '
dF =(drn) dt2= -w2 z drn sin wt
This will cause a torque about the hinge axis of an
amount
dTy_y= (x) (dF) = -w2 x z drn sin wt
or summing up the torques produced by each element of
mass, then the total torque about the hinge axis will be
Tv- y= J - w2 x z drn sin wt
= J x z drn(- w2 sin wt)
Now it is possible to express z in the form
z=a+o0y+ c y2 + ....
where a, 00, c, . .. . are numerical constants chosen to
make the expression for z satisfy the deflection curve.
Substituting this expression for z, then
T y- y= - w2 sin wt f x(a+ o0 y+c y2+ . . . . )drn
= - w2 sin wtJ (ax drn+ o0 x y drn+cxy2drn+ .. .. )
= - w2 00 sin wt(l;x, M + K 1+foJx 1J2drn+ ... . )
But from formula (6),
-(1
K2= K,+M X,Oosince x0= 0
Consequently
Tyy=-00 w2 sin wt(K,+tf xy2 dm+ . ... )
If we let Kg=effective product of inertia of t he aileron,
then Ke= K,+foJ xy2 drn + ... . ____________ (11)
Thus the effective product of inertia of the aileron is
larger than its product of inertia with r espect to the
axis through point 2 by an amount depending on the
deviation of the deflection curve of the a ileron from a
straight line. It should be noticed that point 2 is
located by the intersection between the neutral axis
under no load and the tangent to the deflection curve
of the neutral axis at the inboard edge of the aileron
with the load put on the wing. Calculations carried
out for a large cantilever all-metal wing, using au
expression of the form z= a+ o0 y+ c y2 to express the
deflection curve of the aileron to an accuracy of 1
percent at a ll points, indicated that the terrn!3...Jx y2dm
Oo
was not negligible. Consequently it appears that for a
conservative estimate, the axis of oscilJation of the
wing in bending should be taken at the center line of
the airplane for the purpose of calculating the product
of in ertia of the ailerons.
The effect of en gines located along the span, external
bracing, et cet era, on the location of the axis of oscillation
of the wing in bending should be neglected for
conservative calculations. Preliminary experiments
made at the Materiel Division have indicated that ~uch
concentrated masses along the span and bracing do not
appreciably affect the location of the nodes of vibration
at the fundamental frequencies although their effect
is very marked at the higher frequencies.
Similarly, the product of inertia of the tail control
surfaces should be calculated with respect to their
hinge line and the torsional axis of the rear section of the
fuselage. Substituting the bending axis of the fin or
stabilizer for the torsional axis of the fuselage will lead
to unconservative results as can be seen from the following
discussion.
The product of inertia of control surfaces is not a
quantity inherent merely in the surface itself but is also
a function of the location of the natural axis of oscillation
of the structure by which it is supported. Inasmuch
as this axis of oscillation depends on the mode of
vibration, it is quite obvious that the same control
surface will have different values for its product of
inertia depending on the particular mode of vibration
being investigated. Thus for example, for tail surfaces
of conventional design, a statically unbalanced rudder
will have a considerably smaller product of inertia with
respect to the axis of the fin in bending than it will with
respect to the axis of the fuselage in torsion . Consequently
for conservative results, the axis about which
the rear section of the fuselage twists should be used
in calculating the product of inertia of the rudder.
However, as previously pointed out, if the rear section
of the fuselage has several large cut-outs lying in the
same cross-section, then the product of inertia of the
rudder should also be calculated with respect to its
hinge line and the bending axis of the rear section of the
fuselage.
Similarly, the axis of oscillation of the elevators could
be either the bending axis of the stabilizer or the axis of
symmetry of the airplane. At first glance, it might
appear that the product of ipertia of the elevators with
respect to their hinge line and the axis of symmetry of
t he a irplane would be equal to zero inasmuch as the axis
of symmetry of the airplane is usually an axis of symmetry
of the elevators as well. However, this assumes
that the interconnection of the elevators is extremely
rigid. With the usual type of construction where this
interconnecti ng member consists merely of a tube, more
or less reinforced, tests at the Division have shown th,tt
this assumption of rigidity is not justified. Consequently,
for a conservative estimate, the product of
inert ia of each elevator should be calculatd with respect
to its hinge line and the axis of symmetry of the airplane.
If, in addition, the rear section of the fuselage
is long and has a low flexural rigidity about a horizontal
axis, then the product of inertia of both elevators should
be calculated with respect to the hinge line and the
flexural axis.
THE EXPERIMENTAL DETERMINATION OF THE PRODUCT
OF INERTIA
A means of experimentally determining the product
of inertia of con.trol surfaces is particularly desirable
since it would eliminate long and tedious calculations,
and in addition afford a means of checking t.hat all the
balance weights had actually been in stalled and that the
physical article conformed to the drawings. The three
most promising methods for measuring the product of
inertia of control surfaces will now be discussed.
One meth od for determining the product of inertia
experimentally depends on the formula (see fig. 8).
I o-o= I x-x cos2 a+ I y-y sin2 a - 2 K xu sin a cos a _(12)
The control surface is oscillated about each of the
axes x-x, y-y, which are mutually perpendicular, and
then about third axis o-o lying in the XY plane and
making an angle a with axis X - X. The surface may be
swung as a compound pendulum or else a spring may
be used to provide the restoring force, t he frequency of
the oscillations being measured by means of a stopwatch.
If the control surface is allowed to oscillate as a compound
pendulum under the· influence of gravity, t hen
I =· Wr
4
1r2p- -- - -- -- - __ -- _____ (13)
where
I =moment of inertia about the axis of oscillation
in slug ft2
f = frequency of t he oscillation in cycles per
second
W=weight of t he body in pounds
r=distance from the axis of oscillation to the
center of gravity of the body in feet
5
Having thus determined I x-x, I y-y, and I o-o, their values
are then substituted into formula (1 2) and the value of
Kxu calculated.
This method is the one most commonly used for
measuring the product of inertia experimentally on
account of the simplicity of the test equipment required.
The usual procedure, however, is open to several
serious criticisms.
The control surfaces of modern airplanes are generally
of high aspect ratio, and in addition are approximately
statically balanced about the hinge line. As a result,
it is found that I x-x is large relative to I y-y and that
with the usual location of the axes K xu is small compared
to I x-x and I o-o.
Thus the determination of K by means of this method
introduces the following errors.
First, due to the fact that I depends on f2, then any
experimental error made in determining the true value
of f will cause an error of twice that proportion in calulating
I.
This can be proved as follows from formula 13
Wr
I= 41r2f2
Taking df as equal to /'J. .f and considering that for
small changes,
dl = l'J.l, then
l'J.l= Wrc-2 l'J.f)
4,r2 f3
Dividing this expression by formula 13, then
l'J.l=-20.
I f
Secondly, the result depends on the small difference
between large quantities in as much as in using the
usual location of the axes, Kxu is small compared to
either I x-x cos2a or I o-0 .
In the third place, it is difficult to introd uce the effects
of dihedral, taper in depth, et cetera, into the experimental
set-up.
F inally, several additional fittings must be added
near the leading and trailing edges of the control surface
to provide for its suspension about the chosen axes of
oscillation.
A typical example of the possible errors produced by
the exp.erimental method occurred in the case of the
aileron of a certain pursuit airplane recently submitted
If the restoring force is supplied by a spring instead of to t he Materiel Division for evaluation. Tests carried
gravity, then out by the manufacturer before balance weights were
I=
4
:~- -- --------------- (14) installed showed t hat,
where
k=stiffness of the spring in pounds per foot deflection
d=distance from the axis of oscillation to the
p oint of attachment of the spring to the cont
rol surface, in feet,
l x-x = l 3.83 slug ft2
/y-y = 0.0364 slug ft2
I o-o = 7.42 slug ft2
. !)C= 135.3°
6
Substituting in formula 12, then
I o-o= I x-x cos2 135.3° + I Y-Y sin2 135.3°-2Kx" sin 135.3°
cos 135.3°
7.42= (13:83) ( - 0.7108) 2+ 0.0364 (0.7034)2-2IC.
(0.7034) (- 0.7108)
7.42= 6.99 + 0.02 + 1c.
Kxy= 0.41 slug ft2
This corresponded to a dynamic balance coefficient,
before the balance weights were installed, of 0.323 which
is more than six times the value of 0.05, the maximum
allowed by the Handbook of Instructions for Airplane
Designers, 7th edition, page 306b.
Experience with oscillat ion t ests has indicated that
errors of ± 3 percent in t he value of the moment of
inertia are possible with the usual experimental technique
which includes the use of a stop-watch counting
approximately 50 oscillations at a low frequency. Using
this value as the error in the tests described above,
and assuming an error of - 3 per cent in the values of
I x-x and J y-y and an error of + 3 percent in the value
of I o-o t hen possible values of t he moments of inertia
as measured experimentally could be
I x-x= (13.83) (0. 97) = 13.41 slug ft.2
/ y-y=(0.0364)(0.97)=0.0353 slug ft .2
I o-o= (7.42) (1.03) = 7.64 slug ft.2
Substituting these values in formula (12), then
7.64= (13.41) (- 0.7108)2+ (0.0353) (0.7034)2-2Kxu
(0. 7034) ( - 0. 7108)
7.64= 6.78+0.02 + Kxv
Kx"=0.84 slug ft.2
or a maxi. mum poss1" bl e error o f 0.804.4-l0 .41 105 percent
Thus t his example of an actual aileron with a large
dynamic unbalance shows that an error of ± 3 percent
in the determination of the moments of inertia would
allow a maximum possible error of over 100 percent in
the calculation of the product of inertia. Due to the
geometri cal and mass similarity of corresponding control
s urfaces of most airplanes, it seems likely that the
above example is not unusual, and that in the case of
control surfaces with smaller amounts of dynamic unbalance,
the maximum possible error might be even
greater.
Inasmuch as this method on account of its simJJlicity,
will probably continue to be the most common one used,
it is worth-while studying which value of a should be
used so as to obtain the maximum accuracy. It is safe
to assume that this condition will exist when the effect
of the small difference between large quantities is reduced
to a minimum, that is, when t he ratio between
the sum of the quantities themselves and their differences
is a minimum. Stating this mathematically
then
However, the numerator of this fraction , from equat
ion 12, page 5, is equal to 2 Kxu sin a cos a + 2I o-o
and the denominator is equal to 2Kx" sin a cos a . Consequently
the expression can be rewritten in the form
so that
2Kxu sin a cos a + 2Io-o
2Kxu sin a cos a
l + I o-o
K x" sin a cos a
l o-o
sin a cos
minimum
minimum
minimum
Substituting for I o-o from equation 12, then
I x-x cos2 a+ I Y-_Y sin2a -2K xu sin a cos a = minimu
sm a cos a
I x-x cot a + I v-y tan a - 2Kxy=minimum
l x-x cot a + Jy-y tan a = minimum
I x-x+ I y-ytan2 a ..
tan a =m1n1mum
To solve this, set .i.(lx-x+ Iv-Y tan
2
") = O_( l 6)
da tan a
tan al yy 2 tan a sec2 a - (I x-x + I y-y tan2 a) sec2 a = O
tan2 a
- I x-x sec2 a+ I y-y tan2 a sec2 a
tan2 a
0
1
(- lx-x + l y-y tan2 a)sin2 a = 0
- Ix-x+ ly-y tan2 a= O
a = tan- l"'~Jx-x ____________ (17)
,, y-y
Thus in the example shown previously, Ix-x= 13.83
slug ft.2 and I y-y= 0.0364 slug ft .2, so that
- t - 1 / 13.83
a - an , 0.0364
a= 87° approximately
and the corresponding value of I o-o would be
Uo-o) a-s7o= (13.83) (0.0523)2+ (0.0364) (0.999)2
- (0.41) (0.523) (0.999)
= 0.0378 + 0.0363-0.0428
= 0.0313 slug ft .2
If the same percentage errors are assumed as in the
previous case, then for the maximum possible error
in K xu,
l x-x cos2 a might be (0.0378) (1.03) = 0.0389 slug ft .2
I y-y sin2 a might be (0.0363) (1.03) = 0.0374 slug ft .2
I o-o might be (0.0313) (0.97) = 0.0304 slug ft .2
so that
2 K xu (0.0523) (0.999) = 0.0389+0.0374-0.0304
0.0459
K x" = (2) (0.0523) (0. 999)
= 0.438 slug ft.2
or a maxi. mum poss1" bl e error o f 0.4308.-4lO0. 410 = 6 .8 per-cent
as compa red with the possible error of 105 percen t
in the preceding case. Consequently it can be seen that
use of formula 17 can change t his entire method from
l x-x cos2 a + J y-y sin2 a + I o-o
l l{- X cos2 a + I y-r sin2 a - l o-q
one of extremely dubious accuracy to one yielding satminimum
__ - (l5) . isfactory results. It shoul~ be no~ed here that the axes
7
should be so chosen that the angle a is not t oo close to
0° or 90° inasmuch as t he value of sin a cos a changes
so rapidly in this rnnge that a small error in the measurement
of a will lead to a large error in ](,". Thus
the recommended t est procedure in using this method
is to oscillate the control surface about axes X- X and
Y- Y, which are mutually perpendicular and whose
locations are chosen for maximum experimental accuracy
and convenience, and det ermine I xx and I y y from
formula 13 or 14. Then calculate the value of a from
formula 17 and oscilla te the surface about the corresponding
axis 0- 0, thus determining I 0 0 . Substituting
these values in formula 12, K xv can be calculated·
Finally, the value of K with respect to the axes desired
can be calculated from formula 6 or 9.
Another method for measuring t he product of inertia
depends on formula 4, that is
T K d2ex-x
Y - y = xy~
The procedure is to oscillate the hinge line of control
surface about an axis corresponding to the X- X axis
of the airplane with a niotion -of definite amplitude and
frequency and to measure Tv-v accurately at the same
time. This is done by restraining the motion of the control
surface about the hinge axi.s Y- Y by means of
suitable springs and measuring the amplitude of this
motion by the use of vibration pick-ups. Thus knowing
the spring i,tiffness and the phys ical dimensions of the
syst em, then the torque about the Y- Y axis exerted by
the control surface can be calculated. Similarly knowing
the amplitude and frequency of the oscillations of
the hinge axis Y- Y about the axis X - X, then azex-x
dt2
can be det ermined from formula 3a. Thus knowing
d2ex-x
Tr- Y and ar· then K xy can be calculated directly
from formula 4. Although this method is far more
complicated from the point of view of the experimental
technique and equipment required than the first
method described, it, however, makes up for this by
its increased accuracy since even if JC" were very small,
d2ex-x dr can be made large enough to produce a Tv- v
of appreciable magnitude. The same experimental
set-up can also be used to study the effects of natural
frequency separation and dynamic balance on the
phase lag of control surfaces having various degrees of
damping.
about the X- X axis and add sufficient balancing
weights to the surface until it would no longer oscillate
about the Y- Y axis. Then the sum of the product of
inertias of the control surface and all the balancing
weights added together would be equal to zero. Consequently
the product of inertia of the control surface
would be equal ,in magnitude to the product of inertia
of the balancing weights required. This latter product
of inertia can be easily calculated by the formula
K = ~rnxy ______ _____ _____ (18)
Apparatus suitable for use with these latter two
methods of experimentally measuring the product of
inertia of control surfaces has already been partly contracted
for and the remainder designed at the Material
Division . It is expected that this apparatus will find
considerable use both in the evaluation tests of air-planes
as well as in obtaining general information of
fundamental importance in the study of airplane flutter.
CONCLUSIONS AND RECOMMENDATIONS
The general significance of the fact that the product
of inertia of a body with respect to a given pair of axes
is not zero means that an angular acceleration of that
body about either one of these axes will fnduce a
torque tending to rotate it about the other axis.
For conservative estimates of the critical flutter
speed, the product of inertia of the aileron and each
elevator should be calculated with respect to the hinge
line of the surface and the axis of symmetry of the
airplane. The use of other axes obtained by any means
other than measurements of the vibration amplitudes
in flight near the critical flutter speed is not conservative.
Similarly the product of inertia of the rudder
should be calculated with respect to its hinge line and
the torsional axis of the rear section of the fuselage.
In the case of a fuselage with several cut-outs lying in
the same cross section, the product of inertia of the
rudder or elevator should also be calculated with respect
to the bending axis of the rear section of the fuselage.
The products of inertia of control surfaces that are
statically balanced, are generally quite small and are
independent of the true location of the axes of
oscillation.
In the case of control surfaces whose products of
inertia depend on two axes that are not mutually perpendicular,
the following rule will be found useful:
considering 1> as the angle between the hinge axis and
the axis of oscillation in that quadrant where the center
of gravity of the surface· is located, then if if> is acute,
neglecting this inclination of the axes will always be
conservative ; if obtuse, the result rnay be unconservative,
especially if the product of inertia is small compared
to the moment of inertia.
The third method is merely a variat ion of the preceding
method. It is based on the fact that if ](," were
equal to zero, then Tr-r would also be equal to zero for
The product of inertia can be determined experimentally
to a satisfactory degree of accuracy by any
of the three methods described herein. It should be
that al- stressed in using the method based on the experimental
determination of the moments of inertia about three
This would mean
though the surface was being forced to oscillate about axes lying in the same plane, two of which are mutually
the axis X- X, it would exhibit no t endency to oscillate perpendicular, that the position of the third axis should
about its hinge axis Y- Y. The procedure in carrying not be chosen at random, but should be calculated as
Ollt th<: test would be to oscillate the control surface shown in the- tlisc,mssion by the US<;) of fo~mula (17),
8
F
y
I
y
R
1
FIGURE !.-Product of inertia of the rudder.
Y,
X,
~o
Xz--------
_1_ _._ ___ X.
z.
Y,
f!GU&E ~.-Transferrini: the value of the product of inertia with respect to mw ~et of axes to allother set of parallel ax0.'j,
9
---- 0
y
I
y
0
FIGURE 3.-Iocli nation of t he axes for a typical fuselage-torsion versus rudder mode of vibration.
y
0'----1----
y
FIGURE 4.- Inclination of the axe§ for a t ypical wing-bend ing versus ai)eroq mode of vibratioq,
10
X y
---------_!J
---------xo-----------1-
X y
FIGURE 5.-Parallel axes for a typical fuselage-bending versus rudder mode of vibration.
Hf----.::1
y x'-----
FIGUI\E 6.-Axes not in the same plane. Case of a fuselage-torsion versus elevator mode of vibration.
11
SECTION A-A
FIGURE 7.-Deflection curve of a typical wing in bending.
y
X
0~
y
FIGURE 8.-The three axes used for the experimental determination of the moments of inertia from which the product
of inertia can be calculated.
0